# Kriging with individual errors

When evaluating a Kriging prediction, it is possible to include a hyperparameter $\lambda$ to account for noise in the data. $\lambda$ can be estimated as a parameter in a maximum likelihood estimation (MLE) of the observed data. One effect is that the Kriging estimators will no longer pass through the sample points, and $\lambda$ can be seen as the variance of the noise of the samples' observed values.

Please correct me if I'm saying something wrong!

This approach is fine, but it assumes all the observables have the same error. What if I knew the error in some (all) of the observables? Would it be possible to manually add the error of the i-th sample $\sigma^2_i$ to the corresponding diagonal entry of the correlation matrix? i.e., use $\Psi_{ij} + \sigma_i^2 \delta_{ij}$ instead of simply $\Psi_{ij}$

If so, how could I calculate the variance of the Kriging estimators? My equation for the variance of the Kriging estimators for the simple $\lambda$ case is:

$$\hat{\sigma}^2 [ 1 + \lambda - \psi^T( \Psi + \lambda \bf{I})^{-1} \psi]$$

I don't understand the derivation of this equation, so I don't know how to change it to accomodate individual errors.

Additionally, I feel using individual errors would affect the calculation of the MLE when determining the hyperparameters: if an outlier has a large error, the resulting MLE distribution should have a lower $\hat{\sigma}$. (Right?)

Any insights would be greatly appreciated!

I will not give you a detailed theoretical answer but rather point you to some references.

The kriging with noisy observations does exist. The parameter you call $\lambda$ is often called "nugget" parameter. It is discussed in Chilès and Delfiner's book Modelling spatial uncertainty, for example.

Kriging has also been extended so that you can add your knowledge about the noise value at each observation. This feature is available in DiceKriging package in R. It is often refered as noisy kriging. Rasmussen and Williams (2006) discuss it in some chapters.

• Contrary to your first assertion, the nugget specifically is not an error parameter. It accounts for small-scale spatial variability, not measurement error.
– whuber
Jun 13 '16 at 13:40
• The nugget parameter is indeed not an error parameter and I do not call it this way. It is a term added to the covariance function to account for noises in the dataset. As written by Pepelyshev (2016): "The nugget effect may represent a measurement error or an effect of random values used inside computer model". See chapter 2 of Rasmussen and Willians, too. It can also be used for regularization purposes.
– Pop
Jun 14 '16 at 11:44
• That's a misinterpretation. The problem is that originally, people combined the measurement error with the small-scale variance and called it the "nugget." See Journel & Huijbregts, 1978, for instance, who very clearly interpret the nugget as small-scale variance. This variance and the "noise" are explicitly separated in some models, such as in Diggle & Ribeiro, Model-Based Geostatistics, 2007. Thus, although it's valid that the nugget often includes measurement error, it is incorrect to equate it with "noise," as you do here. That small-scale variance is not "noise" at all.
– whuber
Jun 14 '16 at 13:18